Combinatorics and Arithmetic for Physics: special days

Quotients of Symmetric Polynomial Rings Deforming the Cohomology of the Grassmannian

2 déc. 2020, 16:30

50m

Orateur

Darij Grinberg (Drexel University, Philadelphia, US / currently Germany)

Description

One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian $\mathrm{G}\mathrm{r}(k,n)$ is a quotient of the ring $S$ of symmetric polynomials in $k$ variables. More precisely, it is the quotient of $S$ by the ideal generated by the k consecutive complete homogeneous symmetric polynomials $h_{n-k},h_{n-k+1},\dots ,h_n$. We deform this quotient, by replacing the ideal by the ideal generated by $h_{n-k}-a_1,h_{n-k+1}-a_2,\dots ,h_n-a_k$ for some $k$ fixed elements $a_1,a_2,\dots ,a_k$ of the base ring. This generalizes both the classical and the quantum cohomology rings of $\mathrm{G}\mathrm{r}(k,n)$. We find three bases for the new quotient, as well as an $S_3$-symmetry of its structure constants, a “rim hook rule” for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the $a_i$ as signed indeterminate), which suggests a geometric or
combinatorial meaning for the quotient.

Documents de présentation

cap2020.pdf

Vidéo